Complex Form of Fourier Series

12 Key excerpts on "Complex Form of Fourier Series"

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  Signals and Systems

  A Primer with MATLAB

  • Matthew N. O. Sadiku, Warsame Hassan Ali(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  time-domain representation, where they are described by linear differential equations It is for these reasons that Fourier analysis is used extensively today in science and engineering This chapter begins with the trigonometric Fourier series and how to determine the coefficients of the series Then, we consider the complex exponential Fourier series We will discuss some properties of Fourier series We will cover circuit analy-sis, filtering, and spectrum analyzers as engineering applications of Fourier series We will finally see how we can use MATLAB ® to plot line spectra 4.2 TRIGONOMETRIC FOURIER SERIES The Fourier series can be represented in three ways, the sine–cosine, amplitude– phase, and complex exponential A periodic signal is one that repeats itself every T s In other words, a continuous time signal x ( t ) satisfies x t x t nT ( ) ( ) = + (41) where n is an integer T is the fundamental period of x ( t )

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  Mathematical Methods in Physics

  Partial Differential Equations, Fourier Series, and Special Functions

  • Victor Henner, Tatyana Belozerova, Kyle Forinash(Authors)
  • 2009(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  1.8 The Complex Form of the Fourier Series It is often useful to present Fourier series in complex form. Let us start with the Fourier expansion of the function f (x) with period 2π , f (x) = a 0 2 + ∞ summationdisplay n =1 (a n cos nx + b n sin nx), (1.54) with a n = 1 π π integraldisplay −π f (x) cos nxdx (n = 0, 1, 2,...), b n = 1 π π integraldisplay −π f (x) sin nxdx (n = 1, 2, 3,...). (1.55) From Euler’s formula we have cos nx = 1 2 ( e inx + e −inx ) , sin nx = 1 2i ( e inx − e −inx ) = i 2 ( e −inx − e inx ) , from which we obtain f (x) = a 0 2 + ∞ summationdisplay n =1 bracketleftbigg 1 2 (a n − b n i ) e inx + 1 2 (a n + b n i ) e −inx bracketrightbigg . 1.8. The Complex Form of the Fourier Series 31 Using the notations c 0 = 1 2 a 0 , c n = 1 2 (a n − b n i ) , c −n = 1 2 (a n + b n i ) , we have f (x) = ∞ summationdisplay n =−∞ c n e inx . (1.56) With the Fourier equations for a n and b n (1.55), it is easy to see that the the coefficients c n can be written as c n = 1 2π π integraldisplay −π f (x)e −inx dx (n = 0, ±1, ±2,...). (1.57) It is clear that for functions with period 2l , Equations (1.56) and (1.57) have the form f (x) = ∞ summationdisplay n =−∞ c n e inπx l (1.58) and c n = 1 2l l integraldisplay −l f (x)e − inπx l dx (n = 0, ±1, ±2,...). (1.59) For periodic functions in time t and processes with a period T , the same formulas can be written as f (t ) = ∞ summationdisplay n =−∞ c n e 2inπt T (1.60) and c n = 1 T T/2 integraldisplay −T/2 f (t )e − 2inπx T dt (n = 0, ±1, ±2,...). (1.61) Several useful properties of these results can be easily verified: 1. Because f (x) is real, c n and c −n are complex conjugates, and we have c −n = c ∗ n (where the ∗ denotes complex conjugate); 2. If f (x) is even, all c n are real; 32 1. Fourier Series 3. If f (x) is odd, c 0 = 0 and all c n are pure imaginary.

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  Advanced Mathematical Analysis and Engineering Mathematics

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  This has the advantage of simplifying many of the formulas involved and providing a formulation for Fourier series that more closely resembles the definition followed here. This passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative frequencies. If θ were measured in seconds then the waves e 2 πiθ and e −2 πiθ would both complete one cycle per second, but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related. There is an close connection between the definition of Fourier series and the Fourier transform for functions ƒ which are zero outside of an interval. For such a function we can calculate its Fourier series on any interval that includes the interval where ƒ is not identically zero. The Fourier transform is also defined for such a function. As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of ƒ begins to look like the inverse Fourier transform. To explain this more precisely, suppose that T is large enough so that the interval [− T /2, T /2] contains the interval on which ƒ is not identically zero. Then the n -th series coefficient c n is given by : Comparing this to the definition of the Fourier transform it follows that since ƒ ( x ) is zero outside [− T /2, T /2]. Thus the Fourier coefficients are just the values of the Fourier transform sampled on a grid of width 1/ T . As T increases the Fourier coefficients more closely represent the Fourier transform of the function.

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  Fourier and Laplace Transforms

  • R. J. Beerends, H. G. ter Morsche, J. C. van den Berg, E. M. van de Vrie(Authors)
  • 2003(Publication Date)
  • Cambridge University Press

    (Publisher)

  Then the complex Fourier coefficients c n of f ( t ) , whenever they exist, are defined by c n = 1 T T / 2 − T / 2 f ( t ) e − in ω 0 t dt for n ∈ Z . (3.9) The complex Fourier coefficients have the prefix ‘complex’ since they’ve been determined using complex exponentials, namely, the time-harmonic signals. This prefix has thus nothing to do with the coefficients being themselves complex or not. Like the Fourier coefficients from definition 3.1, the complex Fourier coefficients from definition 3.3 can also be calculated with an integral over an interval that differs from ( − T / 2 , T / 2 ) , as long as the interval has length T . The mapping defined by (3.9) will also be denoted by the transformation pair f ( t ) ↔ c n . Using the complex Fourier coefficients thus defined, we can now introduce the com-plex Fourier series associated with a periodic function f ( t ) . When c n are the complex Fourier coefficients of the periodic function f ( t ) with DEFINITION 3.4 Complex Fourier series period T and fundamental frequency ω 0 = 2 π/ T , then the complex Fourier series of f ( t ) is defined by ∞ n =−∞ c n e in ω 0 t . (3.10) Hence, for periodic functions for which the complex Fourier coefficients exist, a complex Fourier series exists as well. In chapter 4 it will be proven that for piece-wise smooth functions the Fourier series converges to the function at the points of continuity. In (3.8) the complex Fourier coefficients were derived from the real ones. Con-versely one can derive the coefficients a n and b n from c n using a n = c n + c − n and b n = i ( c n − c − n ). (3.11) Therefore, when determining the Fourier series one has a choice between the real and the complex form. The coefficients can always be expressed in each other using (3.8) and (3.11). From (3.5) and (3.6) it follows that for real periodic functions the coefficients a n and b n assume real values.

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  Discrete Wavelet Transformations

  An Elementary Approach with Applications

  • Patrick J. Van Fleet(Author)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 8 COMPLEX NUMBERS AND FOURIER SERIES Complex numbers and Fourier series play vital roles in digital and signal processing. This chapter begins with an introduction to complex numbers and complex arith- metic. Fourier series are discussed in Section 8.2. The idea is similar to that of Maclaurin series – we rewrite the given function in terms of basic elements. While the family of functions x n , n = 0, 1, . . . serve as building blocks for Maclaurin series, complex exponentials e ikω , k ∈ Z are used to construct Fourier series. Also included in Sec- tion 8.2 are several useful properties obeyed by Fourier series. We connect Fourier series with convolution and filters in Section 8.3. We have constructed (bi)orthogonal wavelet filters/filter pairs in Chapters 4, 5 and 7. The constructions were somewhat ad hoc and in the case of the biorthogonal filter pairs, limiting. Fourier series opens up a completely new and systematic way to look at wavelet filter construction. It is important you master the material in this chapter before using Fourier series to construct (bi)orthogonal wavelet filters/filter pairs in Chapter 9. 321 Discrete Wavelet Transformations: An Elementary Approach With Applications, Patrick J. Van Fleet. c  2019 John Wiley & Sons, Inc. Published 2019 by John Wiley & Sons, Inc. Second Edition. 322 COMPLEX NUMBERS AND FOURIER SERIES 8.1 The Complex Plane and Arithmetic In this section we review complex numbers and some of their basic properties. We will also discuss elementary complex arithmetic, modulus, and conjugates. Let’s start with the imaginary number i = √ -1. It immediately follows that i 2 = ( √ -1) 2 = -1 i 3 = i 2 · i = -i i 4 = i 2 · i 2 = (-1)(-1) = 1. In Problem 8.2 you will compute i n for any nonnegative integer n. We define a complex number to be any number z = a + bi, where a and b are any real numbers. The number a is called the real part of z, and b is called the imaginary part of z.

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  Practical Signals Theory with MATLAB Applications

  • Richard J. Tervo(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Recall that the inner product definition for complex signals requires that the complex conjugate of the second term be used in both the numerator and denominator. C n ¼ Z þT =2 T =2 s ðt Þe j 2πnf 0 t dt Z þT =2 T =2 e þj 2πnf 0 t e j 2πnf 0 t dt ð4:36Þ and the denominator equals T, leaving: C n ¼ 1 T Z þT =2 T =2 s ðt Þe j 2πnf 0 t dt ð4:37Þ The complex Fourier series is a linear combination of orthogonal complex expo- nential terms based on complex components of sines and cosines at integer multiples, or harmonics, of a fundamental frequency f 0 . In addition, a zero-frequency (or DC) component found at C 0 acts to “raise or lower” the waveform with respect to the MAGNITUDE PHASE 1 2 1 2π 1 2π 1 2π 1 2π π Φ 4 π 4 f → f → FIGURE 4.39 The signal s ðt Þ = sinðt Þ þ cosðt Þ = ffiffi 2 p cosðt  π=4Þ, as Magnitude vs. Frequency and as Phase vs. Frequency. A phase change (time shift) of a signal has no effect on its magnitude. 4.15 | Complex Fourier Series Components 143 horizontal axis. Significantly, the presence of C 0 does not alter the shape or overall appearance of a waveform. Although integration of an exponential is generally straightforward, there may be times when it will be easier to break the previous complex exponential into sine and cosine terms before integration, leaving: C n ¼ 1 T Z þT =2 T =2 s ðt Þ½cosð2πnf 0 t Þ  j sinð2πnf 0 t Þdt ð4:38Þ In particular, if the signal s ðt Þ is known to be odd or even, then one of the above integrals will be zero. This approach may simplify the necessary computation. Given the above equation, it can be seen that even or odd signals s ðt Þ will have only real or imaginary components, respectively, in the complex Fourier series. This observation will now be explored. 4.15.1 Real Signals and the Complex Fourier Series Consider a real-valued periodic signal that is to be approximated by the complex Fourier series.

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  Signals and Systems Laboratory with MATLAB

  • Alex Palamides, Anastasia Veloni(Authors)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  5 Fourier Series In this chapter, we introduce a way of analyzing = decomposing a continuous-time signal into frequency components given by sinusoidal signals. This process is crucial in the signal processing fi eld since it reveals the frequency content of a signal and simpli fi es the calculation of a system ’ s output. This analysis is based on the use of Fourier series . Up to this point, all signals were expressed in the time domain. With the use of the Fourier series, a signal is expressed in the frequency domain and sometimes a frequency repre-sentation of a signal reveals more information about the signal than its time domain representation. There are three different and equivalent ways that can be used in order to express a signal into a sum of simple oscillating functions, i.e., into a sum of sines, cosines, or complex exponentials. In this chapter, the symbols n and k are often swapped in order for the code written in the examples to be in accordance with the theoretical mathematical equations. 5.1 Orthogonality of Complex Exponential Signals Suppose that x m ( t ) and x k ( t ) are two complex-valued continuous-time periodic signals with period T . These two signals are orthogonal if their inner product is zero. The inner product of x m ( t ) and x k ( t ) is given by I ¼ ð t 0 þ T t 0 x m ( t ) x k * ( t ) dt , (5 : 1) where x k * ( t ) is the complex conjugate of x k ( t ). If I ¼ 0 for m 6 ¼ k , the signals x m ( t ) and x k ( t ) are orthogonal. Suppose now that x m ( t ) ¼ e jm V 0 t , x k ( t ) ¼ e jk V 0 t , and k , m 2 Z . These two signals are orthogonal as I ¼ ð T 0 e jm V 0 t e jk V 0 t dt ¼ ð T 0 e j ( m k ) V 0 t dt ¼ T , k ¼ m 0, k 6 ¼ m & : (5 : 2) In order to verify the orthogonality of complex exponential signals, consider the signals x ( t ) ¼ e j 3(2 p = T ) t and y ( t ) ¼ e j 5(2 p = T ) t . In this case, the complex conjugate of y ( t ) is y *( t ) ¼ e j 5(2 p = T ) t . 249

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  Communications Engineering

  Essentials for Computer Scientists and Electrical Engineers

  • Richard Chia Tung Lee, Mao-Ching Chiu, Jung-Shan Lin(Authors)
  • 2008(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  From Equation (3-64), we can have the following: x ð t Þ¼ x 0 þ X 1 k ¼ 1 r k cos ð 2 p kf 0 t y k Þ : ð 3-84 Þ From this we can see that Equation (3-84) is identical to Equation (3-40). This is why, although the complex exponential form Fourier series contains imaginary terms, the imaginary terms will finally disappear because X k is a conjugate of X k . Again, as in the case where the Fourier series is in trigonometric form, there are special cases for the exponential form. Case 1: y k ¼ 0. This occurs when b k ¼ 0. In this case, there is a cosine function with frequency kf 0 without any phase shift. Case 2: y k ¼ p 2 . This occurs when a k ¼ 0. In this case, the cosine function with frequency kf 0 becomes a sine function with frequency kf 0 without any phase shift. We may claim the following: If X k contains only a real part, the Fourier series contains cosine functions only. If X k contains only an imaginary part, the Fourier series contains only sine functions. Here the phase shift is p = 2 . If X k contains both real and imaginary parts, the Fourier series contains both cosine and sine functions, or equivalently, cosine functions with phase shifts. 3.4 The Fourier Transform In contrast to the Fourier series representation for periodic signals, the Fourier trans-form is used to represent a continuous-time nonperiodic signal as a superposition of complex sinusoids. The idea that a periodic signal can be represented as a sum of sines and cosines with Fourier series is very powerful. We would like to extend this result to aperiodic signals. Extension of the Fourier series to aperiodic signals can be done by extending the period to infinity. In order to take this approach, we assume that the Fourier series of periodic extension of the nonperiodic signal x ð t Þ exists. The nonperiodic signal x ð t Þ is defined in the interval t 0 t t 0 þ T with T > 0.

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  Real Analysis

  A Constructive Approach Through Interval Arithmetic

  • Mark Bridger(Author)
  • 2019(Publication Date)
  • American Mathematical Society

    (Publisher)

  8. THE COMPLEX NUMBERS AND FOURIER SERIES 8.0 Introduction In studying power series we considered the representation of a function as an in nite linear combination of powers of a variable { , that is, sums of the form i ( { ) = 4 X n =0 v n { n . This has some distinct advantages because the building blocks { n are easily computed and the convergence properties are relatively easy to establish. Many important functions, especially the exponential and trigonometric ones, have these Taylor series representations. On the other hand, only functions that are in nitely di erentiable–and not even all of such functions, as the example of h 1 @{ 2 shows–have Taylor series converging to them. This is a major restriction, which excludes even such nice functions as | { | . In this section we look at a di erent approach: representing functions not in terms of powers of { but in terms of the basic periodic functions sin n{ and cos n{ . We are looking for representations of the form i ( { ) = 4 X n =0 d n cos n{ + e n sin n{ , which are called Fourier series representations. Any function that has such a rep-resentation is periodic (since each of sin n{ and cos n{ is 2 periodic). Conversely, many periodic functions can be represented by such series in the sense of pointwise convergence. Moreover, many of those periodic functions that have a convergent Fourier series representation do not have a Taylor series representation (because they are not su ! ciently di erentiable). Finally, Fourier representations have many nice properties that make them easy to work with. For all these reasons, Fourier theory has turned out to be very important both in applications and in spurring the development of the eld of analysis. We will prove in this chapter that every periodic uniformly di erentiable function has a Fourier series converging to it.

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  Mathematical Methods in Engineering and Physics

  • Gary N. Felder, Kenny M. Felder(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 9 Fourier Series and Transforms Before you read this chapter, you should be able to … ∙ identify the amplitude, frequency, period, and phase of the function A cos(px + ). ∙ represent oscillatory functions with complex exponentials. ∙ represent series, finite or infinite, with ∑ notation. (You don’t need to know about Taylor series to read this chapter, but we compare and contrast them to Fourier series in several places. If you’re not familiar with Taylor series you’ll have to ignore those bits.) After you read this chapter, you should be able to … ∙ use a Fourier series to represent a periodic function as a sum of sines and cosines, or equiv- alently as a sum of complex exponentials. ∙ determine if a set of functions is “orthogonal” and if so find the formula for the coefficients of a series built from this set. ∙ create a Fourier series for a function defined on a limited domain by using an “even exten- sion” to create a “Fourier cosine series,” an “odd extension” to create a “Fourier sine series,” or an extension that is neither even nor odd to create a series with sines and cosines. ∙ use a Fourier transform to represent a non-periodic function as an integral of complex exponentials with different amplitudes. ∙ determine the basic properties of a function from a plot of its Fourier transform, or of a Fourier transform from the plot of the function it represents. ∙ use a discrete Fourier transform to model a function defined by a set of data points. ∙ create a multivariate Fourier series for a function of more than one variable. Complicated behavior can often be modeled as the sum of different sine waves of different frequencies. Think of a car going up and down hills. A rock stuck on one of the tires goes up and down quickly due to the rotation of the tire and also goes up and down more slowly due to the hills. A“Fourierseries”mathematicallyrepresentsafunctionsuchastheheightoftherockover time as a superposition of individual sine waves.

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  Signal Processing

  A Mathematical Approach, Second Edition

  • Charles L. Byrne(Author)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  However, our concern here is largely with problems that arise in remote sensing, such as radar, sonar, tomography, and the like, in which the function f of interest is nonzero only on some finite interval. As we shall see, assuming a periodic extension at the start may not be a good idea. 2.3 Complex Exponential Functions The most important functions in signal processing are the complex ex-ponential functions. Using trigonometric identities it is easy to show that the function h : R → C defined by h ( x ) = cos x + i sin x, has the property h ( x + y ) = h ( x ) h ( y ). Therefore, we write it in exponential form as h ( x ) = c x , for some (necessarily complex) scalar c . With x = 1 we have h (1) = cos 1 + i sin 1 = c. Applying the Taylor series expansion e t = 1 + t + t 2 2! + t 3 3! + ..., for t = i we have e i = cos 1 + i sin 1 . Consequently, we have c = e i and h ( x ) = ( e i ) x = e ix . Because it is simpler to work with exponential functions than with trigono-metric functions, we use the identities cos x = 1 2 ( e ix + e − ix ) , Fourier Series and Fourier Transforms 21 and sin x = 1 2 i ( e ix − e − ix ) to reformulate Fourier series and Fourier transforms in terms of complex exponential functions. In place of Equation (2.1) we have f ( x ) ≈ ∞ n = −∞ c n e i nπ L x , with c n = 1 2 L L − L f ( x ) e − i nπ L x dx. (2.5) If f is a continuous function, with f ( − L ) = f ( L ) (so that it has a contin-uous 2 L -periodic extension), then f is uniquely determined by its Fourier coefficients [101, Theorem 2.4], even though the Fourier series may not converge to f ( x ) for some x . 2.4 Fourier Transforms Suppose now that f is a complex-valued function defined on the whole real line. The Fourier transform of f is the function F : R → C given by F ( γ ) = ∞ −∞ f ( x ) e iγx dx. (2.6) Given F , the Fourier Inversion Formula tells us how to get back to f ( x ): f ( x ) = 1 2 π ∞ −∞ F ( γ ) e − iγx dγ.

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  Introductory Signal Processing

  • Roland Priemer(Author)
  • 1990(Publication Date)
  • WSPC

    (Publisher)

  98 CHAPTER TWO FOURIER SERIES In the previous chapter, we expressed sinusoids as a linear combination of complex expo-nential functions, and thereby we were able to solve for the steady-state response of an LTI system to a sinusoid with the same method that is useful for obtaining the response to any exponential function. In this chapter, we shall extend our effort to express signals in terms of complex exponential functions to any periodic signal. This will provide us with a method for characterizing the nature of a periodic function, which is useful for the analysis of periodic signals. Also, through the superposition principle, we will be able to find the steady-state response of an LTI system to any periodic input signal. SECTION 2-1 Periodic Signals and Properties A function x(t) is a periodic function with periodT 0 >Oif for all f,Jt(f)=Jt(f+ T 0 ). If x(t) is periodic, then x(t) = x(t + rT 0 ) for any positive or negative integer r, and rT 0 is also a period of x(t). Generally, the smallest positive time interval, i.e., r = 1, is used to establish the period of x(t). The period determines a frequency that is called the fundamental frequency ©b or/ 0 of x(t) (CQb = 2nf 0 ) by / 0 = 1/T 0 . The unit for / 0 is a Hz (Hertz), where 1 Hz = 1 cycle/second, so that the units for co& are radians/second (rad/sec). Example 2.1. Consider the function x(r) = Acos(co^ + 8), where A > 0 is the amplitude of jc(f),(Oo > 0, is the frequency in rad/sec of x(t), and 0 in radians is the phase angle of x(t). The units of A are the same as the units for jc(r). Obviously, x(t) is a periodic function, but let us apply the definition for periodicity anyway to determine its period.

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